What are the origins of number representation in the mind? Are there any innate building blocks that contribute to our understanding of mathematics and number, or must everything be learned?

Number is an important domain of human knowledge. Many decisions in life are based on quantitative evidence, sometimes with life or death consequences.

Figure 1: Fight or flight?

By now you probably have come to expect that I'll be arguing that there are several innate "building blocks" of cognition that give rise to more complex mathematics. To start with, what are some of the arguments proposed by the empiricists?
(1) Number knowledge is entirely conceptual - it requires seeing objects as belonging to sets;
(2) Number knowledge is abstract. You need to understand the similarity between 3 people, 3 objects, 3 sounds, 3 smells, 3 dollars, 3 seconds, 3 hours, and 3 years;
(3) It doesn't appear to be cross-culturally universal. Some cultures have more advanced mathematics than others; and
(4) babies and monkeys can't do long division.

Surely, humans have something unique that allows us to do things like multivariate regression and construct geometric proofs, however, but let's start at the beginning. I will hopefully convince you that there is an evolutionarily-ancient non-verbal representational systemthat computes the number of individuals in a set. That knowledge system is available to human adults and infants (even in cultures that don't have a count list), as well as to monkeys, rats, pigeons, and so forth.

In the first study (of human adults) that we'll consider, participants were presented with arrays of dots on a computer screen that were presented only for a fraction of a second. In that time, the participants had to determine if the second array had more or fewer dots than the first array. They controlled or counterbalanced dot size, density, shape, and things like that. How did they do in this task?

Figure 2: Results. They were at chance on trials comparing 32 and 34 dots (center). All other comparisons, adults demonstrated above-chance discrimination.

Here's the important point. If they were counting, then they should have been able to discriminate 32 versus 34 dots just as easily as they were able to discriminate 8 versus 10 dots. Also, if they were counting, it should have taken then longer to count 32 versus 64 dots, than for 8 versus 16 dots. Since the dot arrays were displayed for the same amount of time, and they were able to discriminate both of those sets of numerosities equally well (and perfectly), it follows that they weren't counting.

It turns out that the discriminability of two numerosities depends not on the total number of objects, but on the ratio between the two numerosities. Notice that 32 vs. 64 and 8 vs. 16 have the same 1:2 ratio. This also implies that the mental representations of these numerosities are inexact - they are approximations, not exact numbers.

Figure 3: Where the ratio was 1:2, responses were perfect. Success rate decreased as the ratio decreased. When the ratio was 1:1.1, success was basically at chance, but considerable success at 1:1.15.

Is this limited to the visual domain? In the next experiment, participants saw a similar array of dots, and then they heard a set of sounds. This cross-modal comparison (73% accuracy) was almost as accurate as the visual comparison alone (76% accuracy). The participants were also asked to add sets of objects. They were shown two arrays sequentially, and asked to add mentally approximate the total number of dots on both arrays, and shown a third array. They were asked if the sum of the dots in the first two arrays more or less than the total amount of dots in the third array. This resulted in 72% correct responses. Last, this was done cross-modally. They were shown a dot array, then given a sequence of sounds, and asked to add them. Was the sum of the dots and the sounds more or less than the total amount of dots in a new, third, array? Accuracy was 74%. Accuracy was roughly equal across all four of these conditions, suggesting that numerical representations are abstract. It also means that these approximate abstract representations contribute to our ability to add.

Figure 4: Results. Equal success for each condition.

"But," the empiricist says, "these representations have been mapped onto verbal numerals. Even though you might have an approximate estimation of the number of dots, you still use language. You might think 'that's about 50 dots' or 'it seems like there's around 300 dots.' These people have spent years learning and using formal arithmetic." And in response to the empiricist, the nativist says: "Fine. Have it your way. Bring on the babies and animals."

So we gather a group of six month old babies who don't have language yet. We can't ask babies which array has more dots, so we use a habituation paradigm. We show the babies arrays with the same number of dots (say, 8 dots) until they get bored of it and spend less time looking at it. Then we show them a new array (say, 16 dots). Do they look longer at the new array of 16 dots than at a new array of 10 dots? If so, that means that they discriminate 8 versus 16, but not 8 versus 10.

And so it was. Infants successfully discriminated 8 vs. 16, 16 vs. 32, and 4 vs 8. They failed to discriminate 8 vs. 12, 16 vs. 24, or 4 vs. 6 dots. So infants, too, show a ratio limit, though the critical ratio is higher for infants (1:2) than for adults (1:1.15).

How general is this ability? Does it work for sounds as well? A new group of six-month-old and nine-month-old infants were placed between two speakers, and were habituated to a number of sounds coming from a certain direction. Instead of looking time, the measurement was whether or not the infant turned his or her head to orient toward the source of the sounds. They were familiarized to, for example, 8 sounds or 16 sounds, and then tested with 8 and 16 sounds. The findings from this task were similar to the findings in the dot task.

Let's push the question a bit farther. Approximate numerical representation exists for visual objects as well as auditory sounds. What about for actions?

Infants were habituated to a cartoon in which a rabbit jumped either 4 times or 8 times. The total distance of the movement of the rabbit was equivalent, such that the each of the 4 jumps were twice the distance of each of the 8 jumps. This allowed the rabbit to end up at the same location at the end of each set of jumps - so the infants couldn't rely on physical displacement as a correlate of number of jumps. The findings were the same as the dots and sounds. Representation of number is truly abstract, even in infants.

To recap: Before learning to count or learning arithmetic, infants represent and discriminate large numerosities. These representations are approximate, and subject to a ratio limit. These representations are also abstract, and the same ratio limits applies to objects, sounds, and actions. This capability is present in infancy, though it increases in precision through development.

On to the animals.

In the 1950s and 1960s, Dr. Francis Mechner did a series of conditioning experiments with rats. In one such study, rats were trained to press a lever 4, 8, 12, or 16 times in order to receive a reward. Tension on the lever was controlled, so that the rats couldn't rely on total effort as a correlate of the number of level presses.

Figure 5: Presses on the lever by rats.

These data indicate that rats also have inexact, approximate representations of large numerosities, and that there is also a ratio limit. Accuracy decreases as the target number of lever presses increases.

Some years ago, a group from Harvard investigated this question in cotton-top tamarin monkeys as well. They did the same auditory discrimination experiment with the monkeys as had been done previously with infants. The monkeys were habituated to a sequence of sounds coming from the right or left, and then presented with a new number of sounds. Again, the turn of the head to orient towards the sound was used as an indication of discrimination. Their performance was similar to the 9-month-old infants: they discriminated 4 vs. 6 and 8 vs. 12, but not 4 vs. 5 or 8 vs. 10. They could discriminate 2:3 ratios, but not 4:5 ratios.

How evolutionarily-ancient is this cognitive capacity? Our common ancestor with tamarins is relatively recent, compared to say, with fish. So let's look some Italian fish.

Figure 6: Eastern mosquitofish (Gambusia holbrooki). This one is about 4cm long.

Female mosquitofish like to hang out with groups of other females as protection from sexually harassing males.

So you take a female mosquitofish, and you let her habituate to the fishtank. On either side of a long fishtank there are two additional groups of females of differing sizes, in their own tanks.

Figure 7: Something like this.

Meanwhile, you take a male mosquitofish and you deprive him of any females for a whole week. Sucks to be him. You introduce him into the tank with the female, and its game on. He desperately wants the feel of her cold, wet, slimey, scaley body. He can't wait to make sweet fishy love. He sees her from across the tank. He works up his nerve and says, (channeling his best Walter Matthau impression) "Maria, there may be lots of fish in the sea, but you're the only one I want to mount over my fireplace."

If he makes at least 10 attempts to have his way with the female in the first five minutes, then you record which group of fish the female tries to join. She should prefer to hang out with the larger group of females.

When the two groups of female fish were different according a 1:2 ratio, she always chose the larger group, but when the ratio was 2:3, she chose randomly. Just like the monkeys, and just like the human infants.

Figure 8: Success for 1:2 ratios, but not for 2:3 ratios.

What can we conclude from this series of studies?

Animals and humans spontaneously represent large (greater than 4), abstract, approximate numerosities. Animals, human infants, and human adults, show the same ratio signatures. Adult tamarins are on par with 9-month-old human infants. With age or training, discrimination ability becomes more precise, and the the critical ratio is reduced a bit.

The large number cognitive system is evolutionarily-ancient and non-verbal, and is likely innate.

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Here's one for you. I am a writer with a problem; my brain wants to quantify punctuation, specifically colons, semicolons, and commas. I will begin editing a page using standard rules of grammar, but soon I am using a different system based on a ratio, roughly 1 colon: 3 semicolons: 6 commas. This is not the limit per page, but a ratio for each page. Periods are exempt.

This 'automatic' quantifying of punctuation is a real pain! I have to go over a ms. many times, and while trying to correct the mistakes I am apt to lapse into the ratio system without knowing it.

This powers of two recognition and ability to distinguish doubled up/down numbers brings to mind so called "Peasant multiplication" which was a very common system for multiplication and division which relied only on the ability to multiply and divide by two, and addition and subtraction.

It would seem essential that being able to identify relative numbers and numbers in a group is very basic for animals: a pride of lions must decide whether or not to contest another pride's territory, or discretely move on - do they outnumber us, how many males vs. females and cubs; weight, fitness, health. Even an "inventory" of available prey. What we call mathmatics is embedded in every form and function in nature.

Is the cotton tamarin experiment the one done by Hauser that is under investigation at this time?
This obviously doesn't take anything from the conclusion or from the other studies, but I am not sure if it's one of the studies that were going to be under evaluation for possible "retraction"

1) Almost all the experiments deal with discrimination, so the best case scenario is that they tell us something about the number discrimination process, not about number representation - to this extent - discrimination is sensitive to ratios not pure subtractions - that's the only inference you could make.

2) In the rat study, the rats aren't comparing values (at least as presented here), they are responding to absolute values, and it looks like they do reasonably well. What point is there in saying that they are storing ratios not absolute values? I see the rat study to be showing something completely different from the rest of the studies - it is possibly the only one that could say something about the representation of numbers (as opposed to discrimination).

@10: The idea is in order to discriminate two quantities, you have to mentally represent the magnitude of each quantity. The conclusion made is that discriminations are made on the basis of approximate ratio instead of exact numerosity. The rat study demonstrates, with a different sort of task, that they too represent large numbers approxmiately, not exactly.

yes, discriminations are made with approximate ratios, this does not mean the numbers are stored as ratios.

The rats study was also claimed to have "a ratio limit", I'm not sure I follow this, given that it was about tracking absolute values.

I am not against saying that large number representations are "approximate". It seems reasonable. It is the further claim that somehow they are "ratios" that seems weird - cos the ratios are relevant ONLY for discriminatory purposes where there is a second number for ratios. Otherwise, you couldn't store a number as a ratio.

I'm going to jump in with another anecdote about my "odd" relationship with mathematics. First, I need to mention that I score high on verbal / visual intelligence and have very low mathematical aptitude. As a geology student I faced 2 semesters of calculus based physics and 3 semesters of calculus. Baby stuff, I know, but daunting for me. I decided to take the physics semesters BEFORE the calculus course; it sounds crazy, but it helped.

What is odd is that I figured out how to navigate these "foreign" languages using a visual "code" that I made up, a metalanguage, I guess. I connected a description of problem type (thingy-symbol = thingy) with a check list of how to solve it. Gee whiz, it worked. I had no idea what equations "meant" -

I mention this because I suspect there are other people out there, for whom mathematics is not their "native language" who have had to do something similar.

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